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A Probabilistic Mean Field Limit for the Vlasov-Poisson System for Ions

Megan Griffin-Pickering

Abstract

The Vlasov-Poisson system for ions is a kinetic equation for dilute, unmagnetised plasma. It describes the evolution of the ions in a plasma under the assumption that the electrons are thermalized. Consequently, the Poisson coupling for the electrostatic potential contains an additional exponential nonlinearity not present in the electron Vlasov-Poisson system. The system can be formally derived through a mean field limit from a microscopic system of ions interacting with a thermalized electron distribution. However, it is an open problem to justify this limit rigorously for ions modelled as point charges. Existing results on the derivation of the three-dimensional ionic Vlasov-Poisson system require a truncation of the singularity in the Coulomb interaction at spatial scales of order $N^{- β}$ with $β< 1/15$, which is more restrictive than the available results for the electron Vlasov-Poisson system. In this article, we prove that the Vlasov-Poisson system for ions can be derived from a microscopic system of ions and thermalized electrons with interaction truncated at scale $N^{- β}$ with $β< 1/3$. We develop a generalisation of the probabilistic approach to mean field limits that is applicable to interaction forces defined through a nonlinear coupling with the particle density. The proof is based on a quantitative uniform law of large numbers for convolutions between empirical measures of independent, identically distributed random variables and locally Lipschitz functions.

A Probabilistic Mean Field Limit for the Vlasov-Poisson System for Ions

Abstract

The Vlasov-Poisson system for ions is a kinetic equation for dilute, unmagnetised plasma. It describes the evolution of the ions in a plasma under the assumption that the electrons are thermalized. Consequently, the Poisson coupling for the electrostatic potential contains an additional exponential nonlinearity not present in the electron Vlasov-Poisson system. The system can be formally derived through a mean field limit from a microscopic system of ions interacting with a thermalized electron distribution. However, it is an open problem to justify this limit rigorously for ions modelled as point charges. Existing results on the derivation of the three-dimensional ionic Vlasov-Poisson system require a truncation of the singularity in the Coulomb interaction at spatial scales of order with , which is more restrictive than the available results for the electron Vlasov-Poisson system. In this article, we prove that the Vlasov-Poisson system for ions can be derived from a microscopic system of ions and thermalized electrons with interaction truncated at scale with . We develop a generalisation of the probabilistic approach to mean field limits that is applicable to interaction forces defined through a nonlinear coupling with the particle density. The proof is based on a quantitative uniform law of large numbers for convolutions between empirical measures of independent, identically distributed random variables and locally Lipschitz functions.

Paper Structure

This paper contains 26 sections, 23 theorems, 233 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Massless Electrons Model.
  3. The Mean Field Limit for Vlasov-Poisson and Related Systems.
  4. The Probabilistic Approach to the Mean Field Limit for Nonlinear Coupling.
  5. Main Result
  6. Notation.
  7. Particle Dynamics with Smoothed Interactions
  8. Notation for Particle Systems.
  9. Mean Field Particle Dynamics
  10. Selection of Initial Data
  11. Mean Field Limit for the Ionic Vlasov-Poisson System
  12. Notation: Comparison of $N$-tuples.
  13. Structure of the Paper
  14. Notation.
  15. Analysis of the Potential
  16. ...and 11 more sections

Key Result

Theorem 1.6

Let $f_0 \in \mathcal{P}(\mathbb{T}^d \times \mathbb{R}^d)$ be fixed, and suppose that Assumption hyp:f holds. Then the ion Vlasov-Poisson system eq:vpme has a weak solution $f \in C \left ( [0, T) ; \mathcal{P}(\mathbb{T}^d \times \mathbb{R}^d) \right )$. This solution is unique among weak solution where for any $(x,v) \in \mathbb{T}^d \times \mathbb{R}^d$ the trajectory $(\overline Y(t ; x, v),

Figures (2)

  • Figure 1: Straight line paths in $\mathbb{T}^2$ between two points $x$ and $y$ contained in $\overline{B_r}(0)$. \ref{['fig:largeball']} When $r$ is large, the shortest path (solid line), which has length $|x-y|_{\mathbb{T}^2}$, may pass outside of $\overline{B_r}(0)$. \ref{['fig:smallball']} When $r$ is small, the shortest path is contained in $\overline{B_r}(0)$.
  • Figure 2: The union $\overline{B_r}(x) \cup \overline{B_r}(y)$. If an optimal $z^\ast_y$ lies in $\overline{B_r}(x) \cap \overline{B_r}(y)$ (shaded region), then $h(y) \leq h(x)$. Otherwise, we compare $|\nabla g(z^\ast_y)|$ with $|\nabla g(u^\ast_y)|$, by using a mean value theorem argument along the solid line segment connecting $u^\ast_y$ and $z^\ast_y$, whose length is no greater than $|x-y|_{\mathbb{T}^d}$.

Theorems & Definitions (57)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.6
  • Definition 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • Theorem 1.11
  • ...and 47 more